Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations

Abstract

Direct numerical simulations of heavy particles suspended in a turbulent fluid are performed to study the rate of inter-particle collisions as a function of the turbulence parameters and particle properties. The particle volume fractions are kept small (∼10−4) so that the system is well within the dilute limit. The fluid velocities are updated using a pseudo-spectral algorithm while the particle forces are approximated by Stokes drag. One unique aspect of the present simulations is that the particles have finite volumes (as opposed to point masses) and therefore particle collisions must be accounted for. The collision frequency is monitored over several eddy turnover times. It is found that particles with small Stokes numbers behave similarly to the prediction of Saffman & Turner (1956). On the other hand, particles with very large Stokes numbers have collision frequencies similar to kinetic theory (Abrahamson 1975). For intermediate Stokes numbers, the behaviour is complicated by two effects: (i) particles tend to collect in regions of low vorticity (high strain) due to a centrifugal effect (preferential concentration); (ii) particle pairs are less strongly correlated with each other, resulting in an increase in their relative velocity. Both effects tend to increase collision rates, however the scalings of the two effects are different, leading to the observed complex behaviour. An explanation for the entire range of Stokes numbers can be found by considering the relationship between the collision frequency and two statistical properties of the particle phase: the radial distribution function and the relative velocity probability density function. Statistical analysis of the data, in the context of this relationship, confirms the relationship and provides a quantitative description of how preferential concentration and particle decorrelation ultimately affect the collision frequency.

Droplet growth in warm turbulent clouds

“…However, several studies (e.g. Sundaram and Collins, 1997;Wang et al, 1998b;Chun et al, 2005;Ammar and Reeks, 2009) have shown that this is not the case. The persistence or finite lifetime of turbulent structures can make droplet collision rates very much ratelimited by diffusion in the vicinity of the droplets, as is the case of particle transport in a turbulent boundary layer.…”

“…More generally, 12 can be defined as the volume influx containing droplets of radius a 2 to a droplet of radius a 1 over the collision surface S c about the centre of droplets of radius a 1 and radius r c . For spherical droplets N 12 is given by (Sundaram and Collins, 1997;Wang et al, 1998aWang et al, , 2000Ayala et al, 2008b) N 12 = 4π r 2 c n 1 (a 1 )n 2 (a 1 + r c ) (−w 12 )| w 12 <0 ≈ 4π r 2 c 1 2 n 1 (a 1 )n 2 (a 1 + r c ) |w 12 | ≈ 2π r 2 c n 1 (a 1 )n 2 (a 1 + r c ) |w 12 | ≈ 2π r 2 c n 1 n 2 g 12 (r c )|w 12 |,…”

“…Sundaram and Collins, 1997;Reade and Collins, 2000) though with relatively few studies of sedimenting droplets. These studies indicated that particle clustering reaches a maximum for St ∼ 1 and tends to zero (uniform mixing) as St → ∞.…”

“…Direct numerical simulations (DNS) of incompressible turbulence have provided a clear demonstration of the clustering of inertial particles when the particles respond to changes in the turbulent flow on the same time-scale as the smallest eddies change (e.g. Squires and Eaton, 1990;Wang and Maxey, 1993;Sundaram and Collins, 1997). Thus collision rates in a turbulent flow may be enhanced relative to a quiescent one.…”

Turbulent thermal convection driven by heated inertial particles

“…The resulting clusters present a wide range of scales, as illustrated in figure 1. More qualitatively, we provide in figure 6 the radial distribution function (Sundaram & Collins 1997) for the various Stokes numbers and for γ = 570 and C = 0.19. The radial distribution function is defined as the ratio of the number of particle pairs at a given separation r to the expected number of particle pairs for uniformly distributed particles.…”